I saw from a text "From put-call parity, call and put with the same inputs have the same gamma", but I don't see how put-call parity is related to Gamma. Can someone explain? Thanks!
1 Answer
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Put-call parity says that a call and put (worth $C$ and $P$ respectively) with the same strike $K$ have the following relationship with the spot rate $S$, risk-free rate $r$, and time to maturity $T$ --
$$C - P = S - e^{-rT} K$$
Taking the first derivative with respect to $S$,
$$ \frac{\partial C}{\partial S} - \frac{\partial P}{\partial S} = 1 $$
which relates the delta of the call and put. Taking the second derivative,
$$ \frac{\partial^2 C}{\partial S^2} - \frac{\partial^2 P}{\partial S^2} = 0 $$
which implies that the call and put have the same gamma.
